There are languages with a 4th person pronoun. The 3rd person is kind of the main character and the 4th someone else. That helps to disambiguate sentences like "The criminal shot the cop and drove away on his (own or the cop's) bike".
Or the "gay fanfiction problem": "He looked at him and lay his hands on his lap". Is it a happy ending or a sad one? That's one theory why gender in pronouns is so resilient: more often than not, the gendered pronoun can disambiguate which person is talked about. It doesn't always work, a 3rd/4rd person distinction is superior.
You can have an alternate third person pronoun I suppose in order to distinguish two third person individuals, but that doesn't mean there's a fourth person pronoun. The general definition is:
first person - the speaker
second person - the audience, whether present or not present
third person - someone or something other than the audience
So things like "chat" and "breaking the fourth wall" are second person pronouns. There is no fourth person pronoun, because anything other than first and second is covered under third person.
I see why you would analyze it that way but I also see that it deserves a term in its own rights. As you said, it's all terminology. There are no objective definitions, at least not in linguistics